Here is the question:
You want to sell a painting you inherited from a grandparent.
There is a 10% chance it is painted by a famous artist, in which case it's worth 100k.
There is a 30% chance it is painted by an art student, in which case it's worth 2k.
Otherwise, your grandparent must have painted it at a wine and paint night, in which case it's worth $0.
What is the fair price to charge, right now, for a contract that entitles someone to the right, but not the obligation to pay 10k for the painting after we discover who painted it?
My attempt:
So the expected value of the painting is easy to calculate which is 10,600. But how do I use this information to price the contract? Should I price the contract slightly below the expected value to account for the risk involved? If so how much do I discount?
Let us think like this :
We have $100$ Paintings in our Custody.
We know $10\%$ are worth $100k$ , $30\%$ are worth $2k$ , $60\%$ are worth $0k$.
Let $X$ be the Contract Value which the Contract Agent CA will make , to get the fair agreement.
When Painting1 is known to be worth $0k$ , CA is not going to take it for $10k$.
When that Painting is known to be worth $2k$ , CA is not going to take it for $10k$.
When that Painting is known to be worth $100k$ , CA is going to hastily take it for $10k$.
Do this for all Paintings , Painting1 , Painting3 , ....
Eventually , we would have got $100X$ via Contract , $X$ for each Painting.
We would have got $100k$ via transaction , via $10$ worthy Paintings for $10k$.
Buyer would have got $1000k$ total , via $100k$ for $10$ worthy Painting.
Hence :
Our total in-come transaction will be $10\times10k+100X$
Our total out-go transaction will be $10\times100k$
This must be Equal , no loss , no gain :
$10\times10k+100X = 10\times100k$
That gives $100X=1000k-100k$
Hence $X=9k$ is the fair agreement.
Charge less , we will lose.
Charge more , we will gain.
Whether we should make it higher , that is outside the scope : Art Market Experts will have guidelines on the markup.
Buyer may even want to take the Art Student Paintings at a lower Cost , that too is outside the scope : Art Market Experts will negotiate on that.
Purely on Mathematical thinking , $X=9k$ is the fair agreement.